Do you have access to Ruel V. Churchill's Operational Mathematics, 2nd Ed.? If so, try pages 176-181. That may be what you're looking for. If you don't have access, let me know.
Dear MHF Residents,
I need the proof of the Laplace inversion formula
where is a sufficiently large real number, without using the Fourier transform.
Can you please help me in this direction?